Is there a solution for $2^{n-1}\equiv 2^{16}+1\mod n$ or $2^{n-1}\equiv 2^{26}+1\mod n$?

Question

Related to this question : Can I find all solutions of $2^{n-1}\equiv k\mod n$?

Does one of the congruences $$2^{n-1}\equiv 2^{16}+1\mod n$$ and $$2^{n-1}\equiv 2^{26}+1\mod n$$ have an integer solution $n>1$ ?

Enzo Creti checked the second congruences upto $\ 127\cdot 10^9\ $. No solution was found.

The first congruence has no solution below $10^9$

Answer 1

A couple of solutions to $2^{n-1}\equiv 2^{26}+1\pmod{n}$: 567621107057 and 804008475239.